DOI QR코드

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CHARACTERIZATION OF TEMPERED EXPONENTIAL DICHOTOMIES

  • Barreira, Luis (Departamento de Matematica Instituto Superior Tecnico Universidade de Lisboa) ;
  • Rijo, Joao (Departamento de Matematica Instituto Superior Tecnico Universidade de Lisboa) ;
  • Valls, Claudia (Departamento de Matematica Instituto Superior Tecnico Universidade de Lisboa)
  • 투고 : 2018.12.30
  • 심사 : 2019.06.12
  • 발행 : 2019.12.30

초록

For a nonautonomous dynamics defined by a sequence of bounded linear operators on a Banach space, we give a characterization of the existence of an exponential dichotomy with respect to a sequence of norms in terms of the invertibility of a certain linear operator between general admissible spaces. This notion of an exponential dichotomy contains as very special cases the notions of uniform, nonuniform and tempered exponential dichotomies. As applications, we detail the consequences of our results for the class of tempered exponential dichotomies, which are ubiquitous in the context of ergodic theory, and we show that the notion of an exponential dichotomy under sufficiently small parameterized perturbations persists and that their stable and unstable spaces are as regular as the perturbation.

과제정보

연구 과제 주관 기관 : FCT

참고문헌

  1. L. Barreira and Y. B. Pesin, Lyapunov Exponents and Smooth Ergodic Theory, University Lecture Series, 23, American Mathematical Society, Providence, RI, 2002.
  2. L. Barreira and C. Valls, Lyapunov functions for trichotomies with growth rates, J. Differential Equations 248 (2010), no. 1, 151-183. https://doi.org/10.1016/j.jde.2009.07.001 https://doi.org/10.1016/j.jde.2009.07.001
  3. L. Barreira and C. Valls, Smooth robustness of exponential dichotomies, Proc. Amer. Math. Soc. 139 (2011), no. 3, 999-1012. https://doi.org/10.1090/S0002-9939-2010-10531-2 https://doi.org/10.1090/S0002-9939-2010-10531-2
  4. C. V. Coman and J. J. Schaer, Dichotomies for linear difference equations, Math. Ann. 172 (1967), 139-166. https://doi.org/10.1007/BF01350095 https://doi.org/10.1007/BF01350095
  5. Ju. Dalec'kii and M. Krein, Stability of solutions of differential equations in Banach space, translated from the Russian by S. Smith, American Mathematical Society, Providence, RI, 1974.
  6. B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, translated from the Russian by L. W. Longdon, Cambridge University Press, Cambridge, 1982.
  7. T. Li, Die Stabilitatsfrage bei Differenzengleichungen, Acta Math. 63 (1934), no. 1, 99-141. https://doi.org/10.1007/BF02547352 https://doi.org/10.1007/BF02547352
  8. J. L. Massera and J. J. Schaer, Linear differential equations and functional analysis. I, Ann. of Math. (2) 67 (1958), 517-573. https://doi.org/10.2307/1969871
  9. J. L. Massera and J. J. Schaer, Linear Differential Equations and Function Spaces, Pure and Applied Mathematics, Vol. 21, Academic Press, New York, 1966.
  10. O. Perron, Die Stabilitatsfrage bei Differentialgleichungen, Math. Z. 32 (1930), no. 1, 703-728. https://doi.org/10.1007/BF01194662 https://doi.org/10.1007/BF01194662
  11. P. Preda, A. Pogan, and C. Preda, Discrete admissibility and exponential dichotomy for evolution families, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 12 (2005), no. 5, 621-631.